Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Introduction to Binomial Trees Chapter 11

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull A Simple Binomial Model A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option ( Figure 11.1, page 248) A 3-month call option on the stock has a strike price of 21.

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Consider the Portfolio:long  shares short 1 call option Portfolio is riskless when 22  – 1 = 18  or  =  – 1 18  Setting Up a Riskless Portfolio

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Valuing the Portfolio (Risk-Free Rate is 12%) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22  0.25 – 1 = 4.50 The value of the portfolio today is 4.5e – 0.12  0.25 =

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth The value of the shares is (= 0.25  20 ) The value of the option is therefore (= – )

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Generalization Binomial Formula for pricing option ƒ = [ p ƒ u + (1 – p )ƒ d ]e –rT where

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Risk-Neutral Valuation ƒ = [ p ƒ u + (1 – p )ƒ d ]e -rT The variables p and (1  – p ) can be interpreted as the risk-neutral probabilities of up and down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate Su ƒ u Sd ƒ d SƒSƒ p (1  – p )

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Irrelevance of Stock’s Expected Return When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Original Example Revisited Since p is a risk-neutral probability 20 e 0.12  0.25 = 22p + 18(1 – p ); p = Alternatively, we can use the formula Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 S ƒS ƒ p (1  – p )

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Valuing the Option The value of the option is e –0.12  0.25 [   0] = Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull A Two-Step Example Figure 11.3, page 253 Each time step is 3 months K=21, r=12%

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Valuing a Call Option Figure 11.4, page 253 Value at node B = e –0.12  0.25 (   0) = Value at node A = e –0.12  0.25 (   0) = A B C D E F

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull A Put Option Example; K=52 Figure 11.7, page 256 K = 52,  t = 1yr r = 5% A B C D E F

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull What Happens When an Option is American (Figure 11.8, page 257) A B C D E F

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Delta Delta (  ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock The value of  varies from node to node

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Choosing u and d See Page 258 One way of matching the volatility is to set where  is the volatility and  t is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull The Probability of an Up Move

Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull Recommended Study Problems All Quiz Questions 11.1 –